This paper proposes a radical relational ontology for the foundation of physics, asserting that physical reality is grounded not in particles, fields, or spacetime, but in minimal closed relational structures. We introduce the Minimal Phase Closure Triangle (MPCT) system and logically derive two core equations from eight axioms, thereby providing a unified description of quantum mechanics, gravity, time, and cosmology as emergent phenomena. The framework demonstrates that: (1) Quantum superposition corresponds to critical states of local transitions in the relational network; (2) Gravity and spacetime curvature emerge from inhomogeneous distribution of relational density; (3) Time arises as an intrinsic rhythm from the relaxation of this density; and (4) Cosmic accelerated expansion is the macroscopic manifestation of the natural relaxation of the global relational network. Our central result shows that, in the appropriate continuum limits, this framework uniquely yields both the Schrödinger equation and the Einstein field equations, without introducing dark energy, inflatons, or fine-tuning parameters. We provide concrete predictions including a quantum gravity scale ℓ_QG ≈ 30.18 ℓ_P, a gravitational constant G = 1/(16π d_0 σ_d), and falsifiable observational signatures such as particle physics signals around 4 TeV and gravitational wave modulation at 0.125 Hz. This work presents a complete theoretical prototype for a singularity-free, boundaryless cosmology based on relational evolution.